Multiplying two inequalities

Yes your proof is correct. Excellent work reducing the question about multiplying inequalities to a more familiar one of adding inequalities. The only thing I would mention is that taking logarithms and exponentiating are monotone increasing operations. If they were monotone decreasing, the inequalities would flip.

In generally, if you have

$$a \leq x \leq b$$

$$c \leq y \leq d$$

with $a,b, c, d \in \mathbb{R}$ you have that

$$xy \in [\min(ac, ad, bc), \max(bd, ad, cb)]$$